In a large mixing bowl, mix together flour, cream of tartar, salt, baking powder, and baking soda. Mix well until fully combined and set aside.
In a separate large mixing bowl, cream together the butter and sugars using an electric hand or stand mixer. Mix until soft.
Add in eggs, vegetable oil and vanilla. Mix again until fully blended.
Add dry ingredients to the bowl and blend until well combined. The mixture will be very crumbly at first.
Cover the bowl and place inside the refrigerator for 30 minutes to chill.
When chill time is complete, preheat the oven to 350°F and prepare a large baking sheet with parchment paper.
Using a ¼ measuring cup, scoop the mixture tightly from the bowl and roll into round, smooth balls using the palms of your hands. They will be big. Place on baking sheet and continue for the remaining dough.
Bake cookies for 14 to 16 minutes.
Allow the cookies to cool on the baking sheet for 2 minutes before transferring to a cooling rack.
While the cookies are cooling, prepare your frosting. Add butter, powdered sugar, white chocolate, milk, and vanilla to a medium mixing bowl. Blend together for 2-3 minutes until the frosting is smooth and thick. Add ½ cup more of powdered sugar if the frosting is quick to fall off a spoon.
When the cookies have cooled completely, spread the frosting over top, covering the entire cookie except for about ½ inch around the outer edge.
Immediately top with cookie dough bites and mini chocolate chips.
Allow the frosting to cool and stiffen before serving.